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**CBSE Class 12 Mathematics Important Questions Chapter 1 Relations and Functions**

**1 Mark Questions**

**1. A Relation R:A****Ã ****A is said to be Reflexive if â€”â€”â€” for every aÂ
A where A is non**

**empty set.**

**Ans:Â **(a, a)Â R

**2. A Relation R:A****Ã ****A is said to be Symmetric if â€”â€”â€”-Â
a,b,
A**

**Ans:**Â (a, b)Â R, (b, a)Â R

**3. A Relation R:A****Ã ****A is said to be Transitive if â€”â€”â€”â€”-Â
a,b,cÂ
A**

**Ans:**Â (a, b)R, and (b, c)RÂ (a, c)Â R.

**4. Define universal relation? Give example.**

**Ans:**Â A Relation R in a set A called universal relation if each element of A is related to every element of A. Ex. Let = {2,3,4}

R = (AA) = {(2,2),(2,3) (2,4) (3,2) (3,3) (3,4) (4,2) (4,3) (4,4) }

**5. What is trivial relation?**

**Ans:**Â Both the empty relation and the universal relation are some time called trivial relation.

**6. Prove that the function f: RÂ ****Ã ****Â R, given by f(x) = 2x, is one â€“ one.**

**Ans:Â **f is one â€“ one as f(x_{1}) = f (x_{1})

2x_{1Â }= 2x_{2}

x_{1}Â = x_{2}

Prove.

**7. State whether the function is one â€“ one, onto or bijective f: RÂ ****Ã ****Â R defined by f(x) = 1+ x ^{2}**

**Ans:**Â Let x

_{1}, x

_{2}Â x

If f(x

_{1}) = f(x

_{2})

Hence not one â€“ one

**8. Let S = {1, 2, 3}**

**Determine whether the function f: SÂ ****Ã ****Â S defined as below have inverse.**

**f = {(1, 2), (2, 1), (3, 1)}**

**Ans:**Â f(2) = 1 f(3) = 1,

f is not one â€“ one, So that that f is not invertible.

**9. Find gof f(x) = |x|, g(x) = |5x + 1|**

**Ans:**Â gof (x) = g [f(x)]

= g [(x)]

=Â

**10. Let f, g and h be function from R to R show that (f + g) oh = foh + goh**

**Ans:**Â L.H.S = (f + g) oh

= {(f + g) oh} (x)

= (f + g) h (x)

= f [h (x)] + g [h (x)]

= foh + goh

**9. If a * b = a + 3b ^{2}, then find 2 * 4**

**Ans:**Â 2 * 4 = 2 + 3 (4)

^{2}

= 2 + 3Â 16

= 2 + 48

= 50

**11. Show that function f: NÂ ****Ã ****Â N, given by f(x) = 2x, is one â€“ one.**

Ans: the function f is one â€“ one, for

f(x_{1}) = f(x_{2})

2x_{1}Â = 2x_{2}

x_{1}Â = x_{2}

**12. State whether the function is one â€“ one, onto or bijective f: RÂ ****Ã ****Â R defined by f(x) = 3 â€“ 4x**

**Ans:**Â is x_{1}, x_{2}Â R

f(x_{1}) = f(x_{2})

3 â€“ 4x_{1}Â = 3 â€“ 4x_{2}

x_{1}Â = x_{2}

Hence one â€“ one

Y = 3 â€“ 4x

= y

Hence onto also.

**13. Let S = {1, 2, 3}**

**Determine whether the function f: SÂ ****Ã ****Â S defined as below have inverse.**

**f = {(1, 1), (2, 2), (3, 3)}**

**Ans:**Â f is one â€“ one and onto, so that f is invertible with inverse f^{-1}Â = {(1, 1) (2, 2) (3, 3)}

**14. Find got f(x) = |x|, g(x) = |5x -2|**

**Ans:**Â fog (x) = f(g x)

= f{|5x â€“ 2|)

= |5x â€“ 2|

**15. Consider f: {1, 2, 3}Â ****Ã ****Â {a, b, c} given by f(1) = a, f(2) = b and f(3) = c find f ^{-1}Â and show**

**that (f**

^{-1})^{-1Â }= f**Ans:**Â f = {(1, a) (2, b) (3, c)}

f

^{-1}Â = { (a, 1) (b, 2) (c, 3)}

(fÂ

^{-1})Â

^{-1}Â = {(1, a) (2, b) (3, c)}

Hence (f

^{-1})

^{-1}Â = f.

**16. If f(x) = x + 7 and g(x) = x â€“ 7,Â
x
R find (fog) (7)**

**Ans:**Â (fog) (x) = f[g(x)]

= f(x â€“ 7)

= x â€“ 7 + 7

= x

(fog) (7) = (7)

**17. What is a bijective function?**

**Ans:**Â A function f: X Ã Y is said to be one â€“ one and onto (bijective), if f is both one â€“ one and onto.

**18 Let f: RÂ ****Ã ****Â R be define as f(x) = x ^{4}Â check whether the given function is one â€“ one onto,**

**or other.**

**Ans:**Â Let x

_{1}, x

_{2}Â R

If f(x

_{1}) = f(x

_{2})

Not one â€“ one

Not onto.

**19 Let S = {1, 2, 3}**

**Determine whether the function f: SÂ ****Ã ****Â S defined as below have inverse.**

**f = {(1, 3) (3, 2) (2, 1)}**

**Ans:**Â f is one â€“ one and onto, Ao that f is invertible with f^{-1}Â = {(3,1) (2, 3) (1, 2)}

**20 Find gof where f(x) = 8x ^{3}, g(x) = x^{1/3}**

**Ans:**Â gof (x) = g[f(x)]

= g (8x

^{3})

=Â

= 2x

**21. Let f, g and h be function from R + R. Show that (f.g) oh = (foh). (goh)**

**Ans:**Â (f. g) oh

(f. g) h (x)

f[h(x)]. g[h(x)]

foh. goh

**21. Let * be a binary operation defined by a * b = 2a + b â€“ 3. find 3 * 4**

**Ans:**Â 3 * 4 = 2 (3) + 4-3 = 7

**22. show that a one â€“ one function f: {1, 2, 3}Â ****Ã ****Â {1, 2, 3} must be onto.**

**Ans:**Â Since f is one â€“ one three element of {1, 2, 3} must be taken to 3 different element of the co â€“ domain {1, 2, 3} under f. hence f has to be onto.

**23. f: RÂ ****Ã ****Â R be defined as f(x) = 3x check whether the function is one â€“ one onto or other**

**Ans:**Â LetÂ

**24. Let S = {1, 2, 3} Determine whether the function f: S Ã **

**Â S defined as below have inverse.**

f = { (1, 2) (2, 1) (3, 1) }

f = { (1, 2) (2, 1) (3, 1) }

**Ans:**f(2) = 1, f(3) =1

f is not one â€“ one so that f is not invertible

Hence no inverse

**25. Find fog f(x) = 8x ^{3}, g(x) = x^{1/3}**

**Ans:**Â fog (x) = f(gx)

= 8x

**26. If f: RÂ ****Ã ****Â R be given by f(x) =Â
, find fof (x)**

**Ans:**Â

=Â

= x

**27. If f(x) is an invertible function, find the inverse of f(x) =Â
**

**Ans:**Â Let f(x) = y

**4 Marks Questions**

**1. Let T be the set of all triangles in a plane with R a relation in T given by**

**R = {(T1, T2): T1 is congruent to T2}.**

**Show that R is an equivalence relation.**

**Ans.**Â R is reflexive, since everyÂ Â is congruent to itself.

(T1T2)R similarlyÂ (T2T1)Â R

since T1Â T2

(T1T2)Â R, and (T2,T3)Â R

(T1T3)R Since three triangles are

congruent to each other.

**2. Show that the relation R in the set Z of integers given byR={(a, b) : 2 divides a-b}. is equivalence relation.**

**Ans.**Â R is reflexive , as 2 divide a-a = 0

((a,b)R ,(a-b) is divide by 2

Â (b-a) is divide by 2 Hence (b,a)Â R hence symmetric.

Let a,b,cÂ Â Â Z

IfÂ (a,b)Â R

And (b,c)Â Â R

ThenÂ a-b and b-c is divided by 2

Â a-b +b-c is even

Â (a-c is even

Â (a,c)Â R

Hence it is transitive.

**3. Let L be the set of all lines inÂ Â Â plane and R be the relation in L define if R = {(l1, L2 ): L1 isÂ
Â to L2 } . Show that R is symmetric but neither reflexive nor transitive.**

**Ans.**Â R is not reflexive , as a line L1 cannot beÂ Â to itselfÂ i.eÂ (L1,L1 )Â R

Â L1Â L2

L2Â L1

Â (L2,L1)R

L1Â Â L2 and L2Â L3

Then L1 can never beÂ Â Â to L3 in fact L1 || L3

i.e (L1,L2)Â R, (L2,L3)Â Â R.

But (L1, L3)Â R

**4. Check whether the relation R defined in the set {1, 2, 3, 4, 5, 6} asR = {(a, b): b = a+1} is reflexive, symmetric or transitive.**

**Ans.Â **R = {(a,b): b= a+1}

Symmetric or transitive

R = {(1,2) (2,3) (3,4) (4,5) (5,6) }

R is not reflective , because (1,1)Â R

R is not symmetric because (1,2)Â R but (2,1)Â R

(1,2)Â R and (2,3)Â R

But (1,3)Â R Hence it is not transitive

**5. Let L be the set of all lines in xy plane and R be the relation in L define as R = {(L1, L2): L1 || L2} Show then R is on equivalence relation.**

**Find the set of all lines related to the line y=2x+4.**

Ans. L1||L1Â Â Â i.e (L1, L1)Â Â RÂ Hence reflexive

L1||L2 then L2 ||L1Â Â i.e (L1L2)Â R

Â (L2,L)Â R Hence symmetric

We know the

L1||L2Â and L2||L3

Then L1|| L3

Hence Transitive .Â y = 2x+K

**When K is real number.**

**6. Show that the relation in the set R of real no. defined R = {(a, b) : a
Â b3 }, is neither reflexive nor symmetric nor transitive.**

**Ans.**Â **(i)**Â (a, a)Â Â Â Which is false R is not reflexive.

**(ii)**Â Â Which is false R is not symmetric.

**(iii)**Â Â Which is false

**7. Let A = N
N and * be the binary operation on A define by (a, b) * (c, d) = (a + c, b + d)Show that * is commutative and associative.**

**Ans.**Â **(i)**Â (a, b) * (c, d) = (a + c, b + d)

= (c + a, d + b)

= (c, d) * (a, b)

Hence commutative

**(ii)**Â (a, b) * (c, d) * (e, f)

= (a + c, b + d) * (e, f)

= (a + c + e, b + d + f)

= (a, b) * (c + e, d + f)

= (a, b) * (c, d) * (e, f)

Hence associative.

**8. Show that if f:Â
Â is defining by f(x) =Â
and g:Â
is define by**

**g(x) =Â
then fog = IA and gof = IBÂ whenÂ
; IA (x) = x, for all x
A, IB(x) = x, for all x
B are called identify function on set A and B respectively.**

**Ans.**Â gof (x) =Â

Which implies that gof = IB

And Fog = IA

**9. Let f: NÂ Ã ****Â N be defined by f(x) =Â
**

**Examine whether the function f is onto, one â€“ one or bijective**

**Ans.**Â

f is not one â€“ one

1 has two pre images 1 and 2

Hence f is onto

f is not one â€“ one but onto.

**10. Show that the relation R in the set of all books in a library of a collage given by R ={(x, y) : x and yÂ have same no of pages}, is an equivalence relation.**

**Ans.**Â **(i)**Â Â (x, x)Â Â R, as x and x have the same no of pages for all xRÂ Â R is reflexive.

**(ii)**Â Â (x, y) R

x and y have the same no. of pages

y and x have the same no. of pages

(y, x)Â R

(x, y) = (y, x) R is symmetric.

**(iii)**Â Â if (x, y)Â Â R, (y, y)Â Â R

(x, z)Â R

Â R is transitive.

**11. Let * be a binary operation. Given by a * b = a â€“ b + abIs * :**

**(a) Commutative**

**(B) Associative**

**Ans.**Â **(i)**Â Â a * b = a â€“ b + ab

b * a = b â€“ a + ab

a * bÂ Â b * a

**(ii)**Â Â a * (b * c) = a * (b â€“ c + bc)

= a â€“ (b â€“ c + bc) + a. (b â€“ c + bc)

= a â€“ b + c â€“ bc + ab â€“ ac + abc

(a * b) * c = (a â€“ b + ab) * c

= [ (a â€“ b + ab) â€“ c ] + ( a â€“ b + ab)

= a- b + ab â€“ c + ac â€“ bc + abc

a * (b * c)Â Â Â (a * b) * c.

**12. Let f: R Ã ****Â R be f (x) = 2x + 1 and g: RÂ ****Ã ****Â R beÂ g(x) = x2 â€“ 2 find (i) gof (ii) fog**

**Ans.**Â **(i)**Â Â gof (x) = g[f(x)]

= g (2x + 1)

= (2x + 1)2 â€“ 2

**(ii)**Â Â fog (x) = f (fx)

= f (2x + 1)

= 2(2x + 1) + 1

= 4x + 2 + 1 = 4x + 3

**13. Let A = R â€“ {3} and B = R- {1}. Consider the function of f: A Ã ****Â B defined by**

**f(x) =Â
Â is f one â€“ one and onto.**

**Ans.**Â Let x1 x2Â Â A

Such that f(x1) = f(x2)

f is one â€“ one

Hence onto

**14. Show that the relation R defined in the set A of all triangles asR = {
Â is similar to T2 }, is an equivalence relation. Consider three right angle triangles T1 with sides 3, 4, 5. T2 with**

sides 5, 12, 13 and T3 with sides 6, 8, 10. Which triangles among T1, T2 and T3 are related?

**Ans.**Â **(i)**Â Â Each triangle is similar to at well and thus (T1, T1)Â Â R

Â R is reflexive.

**(ii)**Â Â (T1, T2)Â Â R

Â T1 is similar to T2

T2 is similar to T1

(T2, T1)Â Â R

R is symmetric

**(iii)**Â Â T1 is similar to T2 and T2 is similar to T3

T1 is similar to T3

(T1, T3)Â Â R

R is transitive.

Hence R is equivalence

**(II)**Â part Â T1 = 3, 4, 5

T2 = 5, 12, 13

T3 = 6, 8, 10

T1 is relative to T3.

**15. Determine which of the following operation on the set N are associative and which are commutative.**

**(a) a * b = 1 for all a, bÂ
N**

**(B) a * b =Â
Â for all a, b,Â
N**

**Ans.**Â **(a)**Â Â a * b = 1

b * a = 1

for all a, bÂ Â N also

(a * b) * c = 1 * c = 1

a * (b * c) = a * (1) = 1 for all, a, b, c R N

Hence R is both associative and commutative

**(b)**Â Â a * b =Â ,Â b * a =Â

Hence commutative.

(a * b) * c =Â

=

=Â

* is not associative.

**17. Let A and B be two sets. Show that f: AÂ
B Ã ****Â BÂ
A such that f(a, b) = (b, a) is a bijective function.**

**Ans.**Â Let (a1 b1) and (a2, b2)Â Â AÂ Â B

**(i)**Â Â f(a1 b1) = f(a2, b2)

b1 = b2 and a1 = a2

(a1 b1) = (a2, b2)

Then f(a1 b1) = f(a2, b2)

(a1 b1) = (a2, b2) for all

(a1 b1) = (a2, b2)Â Â AÂ Â B

**(ii)**Â Â f is injective,

Let (b, a) be an arbitrary

Element of BÂ Â A. then bÂ Â B and aÂ Â A

(a, b) )Â Â (AÂ Â B)

Thus for all (b, a)Â Â BÂ Â A their exists (a, b) )Â Â (AÂ Â B)

Hence that

f(a, b) = (b, a)

So f: AÂ Â B Ã BÂ Â A

Is an onto function.

Hence bijective

**18. Show that the relation R defined by (a, b) R (c, d)Â
a + b = b + c on the set NÂ
N is an equivalence relation.**

**Ans.**Â (a, b) R (c, d)Â Â a + b = b + c where a, b, c, dÂ Â N

(a, b ) R (a, b)Â Â a + b = b + a (a, b)Â Â NÂ Â N

R is reflexive

(a, b) R (c, d)Â a + b

= b + c (a, b ) (c, d)Â Â NÂ Â N

d + a = c + b

c + b = d + a

Â (c, d) R (a, b) (a, b), (c, d)Â Â NÂ Â N

Hence reflexive.

(a, b) R (c, d)Â Â a + d = b + c Â Â (1) (a, b), (c, d)Â Â NÂ Â N

(c, d) R (e, f)Â Â c + f = d + e Â Â (2) (c, d), (e, f)Â Â NÂ Â N

Adding (1) and (2)

(a + b) + [(+f)] = (b + c) + (d + e)

a + f = b + e

(a, b) R (e, f)

Hence transitive

So equivalence

**19. Let * be the binary operation on H given by a * b = L. C. M of a and b. find**

**(a) 20 * 16**

**(b) IsÂ * commutative**

**(c) Is * associative**

**(d) Find the identity of * in N.**

**Ans. (i)**Â Â 20 * 16 = L. C.M of 20 and 16

= 80Â Â Â Â

**(ii)**Â Â a * b = L.C.M of a and b

= L.C.M of b and a

= b * a

**(iii)**Â Â a * (b * c) = a * (L.C.M of b and c)

= L.C.M of (a and L.C.M of b and c)

= L.C.M of a, b and c

Similarity

(a * b) * c = L. C.M of a, b, and c

**(iv)**Â Â a * 1 = L.C.MÂ of a and 1= a

=1

**20. If the function f: R Ã ****Â R is given by f(x) =Â
Â and g: RÂ ****Ã ****Â R is given by g(x) = 2x â€“ 3, FindÂ **

**(i) fog Â **

**(ii) gof. Is f-1 = g**

**(iii) Â fog = gof = x**

**Ans.**Â **(i)**Â Â fog (x) = f [g(x)]

= f (2x â€“ 3)

=Â

= x

**(ii)**Â Â gof (x) = g [f(x)]

= x

**(iii)**Â Â fog = gof = x

Yes,

**21. Let L be the set of all lines in Xy plane and R be the relation in L define as R = {(L1, L2): L1 || L2} Show then R is on equivalence relation.**

**Find the set of all lines related to the line y=2x+4.**

**Ans.**Â L1||L1Â Â Â i.e (L1, L1)Â Â RÂ Hence reflexive

L1||L2 then L2 ||L1Â Â i.e (L1L2)Â R

Â (L2, L)Â R Hence symmetric

We know the

L1||L2 and L2||L3

Then L1|| L3

Hence Transitive.Â y = 2x+K

When K is real no.